Question

At a given instant, there are 25% undecayed radioactive nuclei in a sample. After 10 s, the number of undecayed nuclei reduces to 12.5%. Calculate (a) mean life of the nuclei and (b) the time in which the number of decayed nuclei will further reduce to 6.25% of the reduced number

Answer 1

Half-life of radioactive sample, i.e., the time in which number of undecayed nuclei becomes half (T) is $10s$.
a. Mean life, $\tau =\frac{T}{lo{g}_{e}2}=\frac{10}{0.693}s=1.443\times 10=14.43s$
b. The reduced number further reduced to 6.25% in n half-lives given by:
$\frac{N}{100}={\left(\frac{1}{2}\right)}^n\Rightarrow \frac{6.25}{100}={\left(\frac{1}{2}\right)}^n$

$\Rightarrow \frac{1}{16}={\left(\frac{1}{2}\right)}^n\Rightarrow {\left(\frac{1}{2}\right)}^4={\left(\frac{1}{2}\right)}^n$

$\Rightarrow n=4$

Time, $t=4T=4\times 10=40s$